2r^2+10r+11=0

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Solution for 2r^2+10r+11=0 equation:


Simplifying
2r2 + 10r + 11 = 0

Reorder the terms:
11 + 10r + 2r2 = 0

Solving
11 + 10r + 2r2 = 0

Solving for variable 'r'.

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
5.5 + 5r + r2 = 0

Move the constant term to the right:

Add '-5.5' to each side of the equation.
5.5 + 5r + -5.5 + r2 = 0 + -5.5

Reorder the terms:
5.5 + -5.5 + 5r + r2 = 0 + -5.5

Combine like terms: 5.5 + -5.5 = 0.0
0.0 + 5r + r2 = 0 + -5.5
5r + r2 = 0 + -5.5

Combine like terms: 0 + -5.5 = -5.5
5r + r2 = -5.5

The r term is 5r.  Take half its coefficient (2.5).
Square it (6.25) and add it to both sides.

Add '6.25' to each side of the equation.
5r + 6.25 + r2 = -5.5 + 6.25

Reorder the terms:
6.25 + 5r + r2 = -5.5 + 6.25

Combine like terms: -5.5 + 6.25 = 0.75
6.25 + 5r + r2 = 0.75

Factor a perfect square on the left side:
(r + 2.5)(r + 2.5) = 0.75

Calculate the square root of the right side: 0.866025404

Break this problem into two subproblems by setting 
(r + 2.5) equal to 0.866025404 and -0.866025404.

Subproblem 1

r + 2.5 = 0.866025404 Simplifying r + 2.5 = 0.866025404 Reorder the terms: 2.5 + r = 0.866025404 Solving 2.5 + r = 0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = 0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = 0.866025404 + -2.5 r = 0.866025404 + -2.5 Combine like terms: 0.866025404 + -2.5 = -1.633974596 r = -1.633974596 Simplifying r = -1.633974596

Subproblem 2

r + 2.5 = -0.866025404 Simplifying r + 2.5 = -0.866025404 Reorder the terms: 2.5 + r = -0.866025404 Solving 2.5 + r = -0.866025404 Solving for variable 'r'. Move all terms containing r to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + r = -0.866025404 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + r = -0.866025404 + -2.5 r = -0.866025404 + -2.5 Combine like terms: -0.866025404 + -2.5 = -3.366025404 r = -3.366025404 Simplifying r = -3.366025404

Solution

The solution to the problem is based on the solutions from the subproblems. r = {-1.633974596, -3.366025404}

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